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References
P. D. Lax, “Integrals of nonlinear equations of evolution and solitary waves,” Comm. Pure Appl. Math.,21, No. 2, 467–490 (1968).
R. M. Miura (editor), Bäcklund Transformations, The Inverse Scattering Method, Solutions, and Their Applications, Springer, Berlin (1976) (Lecture Notes in Math.,515).
C. Rogers and W. F. Shadwick, Bäcklund Transformations and Their Applications, Academic Press, New York (1982).
M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform [Russian translation], Mir, Moscow (1987).
V. E. Zakharov and A. B. Shabat, “An integration scheme for nonlinear equations of mathematical physics by the inverse scattering method,” Funktsional. Anal. i Prilozhen.,8, No. 3, 43–53 (1974).
V. E. Zakharov and A. B. Shabat, “Integration of nonlinear equations of mathematical physics by the inverse scattering method,” Funktsional. Anal. i Prilozhen.,13, No. 3, 13–22 (1979).
V. E. Zakharov, S. V. Manakov, S. P. Novikov, and L. P. Pitaevskii, Theory of Solitons. The Inverse Problem Method [in Russian], Nauka, Moscow (1980).
S. P. Novikov, “A method for solving a periodic problem for the KdV equation and its generalization,” in: Solitons [Russian translation], Mir, Moscow, 1983, pp. 348–362.
F. Calogero and M. C. Nucci, “Lax pairs galore,” J. Math. Phys.,32, No. 1, 72–74 (1991).
R. Hermann, “Prolongations, Bäcklund transformations, and Lie theory as means for studying nonlinear differential equations,” in: Solitons in Action [Russian translation], Mir, Moscow, 1981, pp. 45–71.
I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, “Quasilocal symmetries of equations of mathematical physics,” in: Mathematical Modeling. Nonlinear Differential Equations of Mathematical Physics [in Russian], Nauka, Moscow, 1987, pp. 22–56.
I. Sh. Akhatov, R. K. Gazizov, and N. Kh. Ibragimov, “Quasilocal symmetries of equations of nonlinear heat conduction,” Dokl. Akad. Nauk SSSR,295, No. 1, 75–78 (1987).
G. A. Rudykh and È. I. Semënov, Commutative Representations and Bäcklund Transformations for a Nonlinear Evolution Equation with One Space Variable [Preprint, No. 7] [in Russian], Irkutsk, Irkutsk. Vychisl. Tsentr Sibirsk. Otdel. Akad. Nauk SSSR (1990).
G. A. Rudykh and È. I. Semënov, “A construction of exact solutions to a many-dimensional quasilinear equation of heat conduction,” Zh. Vychisl. Matematiki i Mat. Fiziki,33, No. 8, 1228–1239 (1993).
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Translated from Sibirskii Matematicheskii, Vol. 36, No. 1, pp. 164–176, January–February, 1995.
In conclusion, the authors express their sincere gratitude to L. V. Ovsyannikov, A. P. Chupakhin, and all participants of the theoretical seminar of the M. A. Lavrent'ev Institute of Hydrodynamics for useful criticism and discussion of the authors' preprint [13] whose first two sections are presented in the article and the fourth is published in [14].
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Rudykh, G.A., Semënov, É.I. Lax representations and Bäcklund transformations for one-dimensional nonlinear evolution equations. Sib Math J 36, 147–159 (1995). https://doi.org/10.1007/BF02113929
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DOI: https://doi.org/10.1007/BF02113929